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I wonder if anyone knows a research article that uses the canonical path method (or congestion ratio) to show that a Poincare inequality holds (see the images below) for a continuous-time Markov chain on a countable state space.

The canonical path method is a famous way of showing that the Poincare inequality holds (or the spectral gap is positive), which, in turn, implies the exponential ergodicity of the Markov chain. Here is a snapshot of the method and the relative theorem (from page 47 and 48 of Counting and Markov Chains by Mark Jerrum): enter image description here

enter image description here

All the references I have found regarding the canonical path method assume that the relevant Markov chains operate within a finite state space. However, as evident from the above images, I believe this method can still be applied to continuous-time Markov chains on a countable state space. Therefore, I kindly request information on any papers that have utilized the canonical path method for a continuous-time Markov chain on a countable state space

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I might be wrong, but I think the canonical path method will not work for infinite state space. This is because there could be infinitely many paths with arbitrary lengths going through a particular edge, which means that the Poincare constant will be infinite, which is vacuous.

Intuitively, the Poincare constant will be somewhat dependent on the diameter of the region. For instance, consider a random walk on a path graph with $n$ vertices, we will have the mixing time to be $O(n^2)$, which is the square of the diameter. You can prove this mixing time using the canonical path method and see why.

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  • $\begingroup$ Thank PepeHands. So, when the Poincare constant is infinite for the countable state space case, I can conclude that the canonical path method is silent about showing exponential ergodicity. Do you agree with this? $\endgroup$
    – IamHungry
    Commented Oct 16, 2023 at 22:04
  • $\begingroup$ But what do you think if the Poincare constant is finite for a countable state space (I think this is possible for a simple 1d birth and death model)?? $\endgroup$
    – IamHungry
    Commented Oct 16, 2023 at 22:05
  • $\begingroup$ I agree that Poincare constant can be finite for a countable state space, but the canonical path method probably does not imply that however. $\endgroup$
    – PepeHands
    Commented Oct 17, 2023 at 1:00
  • $\begingroup$ May I ask what setting you are looking into like BDP or queuing systems? Maybe it will give me some intuitions. $\endgroup$
    – PepeHands
    Commented Oct 17, 2023 at 1:02
  • $\begingroup$ I am interested in continuous-time Markov chains on a countable state space such as M/infinite/infinite queueing systems. $\endgroup$
    – IamHungry
    Commented Oct 18, 2023 at 2:11

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